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In modern signal processing, many applications use filter banks to decompose a signal into several narrower frequency ranges, which are convenient for further processing. For example, adaptive filter applications use filter banks to reduce computational complexity and improve the convergence performance by decomposing a high-order adaptive filter into several low-order adaptive filters. Audio processing applications use filter banks for room simulation and digital audio compression. Digital communications applications use filter banks in digital transmultiplexers, channel equalization precoding, and discrete multitone modulation. Before using filter banks, you must have some basic knowledge of multirate filters.
A filter bank has two parts: an analysis bank and a synthesis bank. Each bank is a set of bandpass filters. The filters in the analysis bank are analysis filters and the filters in the synthesis bank are synthesis filters. The following figure shows the structure of a filter bank.
The analysis bank consists of M analysis filters Hk(z), where k represents the kth subband. Uniform filter banks in the LabVIEW Digital Filter Design Toolkit have the property that the number of subbands, M, is equal to the decimation factor of the subband filters. The analysis filters, Hk(z), decompose the input signal, x[n], into multiple subband signals, Vk[n]. Each subband signal carries a specific frequency range of the input signal. The synthesis filters, Fk(z), reconstruct the original signal, y[n], from the processed subband signals, Wk[n].
When designing filter banks, it is important to consider the signal reconstruction and frequency selectivity.
If you do not apply a processing unit to the subband signals, the synthesis bank reconstructs the original signal, perfectly or near-perfectly. You can evaluate the signal reconstruction by examining distortions in the filter bank. There are three fundamental distortions in filter banks: aliasing, magnitude distortion, and phase distortion. If a filter bank is free from these distortions, the filter bank is a perfect reconstruction (PR) filter bank. If these distortions are small, the filter bank is a near-perfect reconstruction (NPR) filter bank. If a filter bank has perfect reconstruction, the output signal is a scaled and delayed version of the input signal.
Frequency selectivity indicates the ability of a filter bank to separate frequency subbands. Each subband contains signal components in a different frequency range. You can evaluate the frequency selectivity by examining the stopband attenuation and the transition bands. To achieve good frequency selectivity, the filter bank must have a high stopband attenuation and narrow transition bands.
It is often difficult to design PR filter banks that achieve good frequency selectivity. However, NPR filter banks can achieve high stopband attenuation and narrow transition bands while introducing only small distortions. Thus, in practice, NPR filter banks often have better frequency selectivity than PR filter banks. Consequently, many real-world applications use NPR filter banks instead of PR filter banks.